This male/ female Fibonacci is another application of the pair type
functions.
In Mathematica:
f[n_]:=(1/(n+1))^Mod[n,2]*(n/(n+1))^(1-Mod[n,2])
but
g[n_]:=(n/(n+1))^Mod[n,2]*(1/(n+1))^(1-Mod[n,2])
doesn't seem to work. I had to change it to:
g[n_]:=If[Mod[n,2]==1,(n/(n+1)),(1/(n+1))]
The modulo power version seems functionallt equivalent,
but fails completely in the zeta function versions of these.
(* (1/(n+1),n/(1+n)) pair function used to get a dual population
Fibonacci *)
(* if the Fibonacci is a rabbit population , thn it has male and femal
components*)
(* in this case the gfib ( female) population is always larger or the same*)
(* natural birth rate has the female popoulation slightly larger than
the male in many mammals*)*)
digits=50
f[n_]:=(1/(n+1))^Mod[n,2]*(n/(n+1))^(1-Mod[n,2])
g[n_]:=If[Mod[n,2]==1,(n/(n+1)),(1/(n+1))]
fib[n_Integer?Positive] :=fib[n] =fib[n-1]+fib[n-2]
fib[0]=0;fib[1] = 1;
ffib[n_Integer?Positive] :=ffib[n] =ffib[n-1]*f[n-1]+ffib[n-2]*f[n-2]
ffib[0]=0;ffib[1] = 1;
gfib[n_Integer?Positive] :=gfib[n] =gfib[n-1]*g[n-1]+gfib[n-2]*g[n-2]
gfib[0]=0;gfib[1] = 1;
a=Table[Floor[ffib[n]*fib[n]],{n,0,digits}]
b=Table[Floor[gfib[n]*fib[n]],{n,0,digits}]
{0,1,0,1,1,3,4,7,11,18,29,47,75,123,197,321,514,836,1343,2181,3508,5692,9167,
14865,23959,38838,62635,101503,163773,265344,428291,693791,1120191,1814345,
2930173,4745365,7665395,12412755,20054413,32471888,52470417,84953526,
137291667,222271983,359249034,581585233,940082660,1521822386,2460102246,
3982297570,6438059697}
{0,1,0,1,2,3,5,8,13,20,34,54,88,141,230,368,599,962,1562,2512,4077,6562,10644,
17149,27804,44827,72655,117201,189907,306473,496500,801528,1298303,2096510,
3395454,5484273,8881231,14347563,23232342,37537787,60778546,98216903,
159015502,256996472,416059948,672493991,1088669150,1759816751,2848763556,
4605344794,7454779663}